Lecture Note Microeconomic Theory 1 - Yonsei

1 Lecture NoteMicroeconomic Theory 1 Basic analytical framework of modern economics: Economic environments: Number of agents, individuals characteristics (preference,technology, endowment), information structures, institutional economic environments Behavioral assumption: Selfish and rational agents Economics institutional arrangements: Economic mechanism Equilibrium Evaluation: Welfare analysis (first-best, second-best) Mathematics as a tool for economists To describe model and assumptions clearly and precisely To make analysis rigorous To obtain results that may not be available through verbal arguments To reduce unnecessary debates1 Chapter 1 Consumer Theory4 Building blocks: consumption set, feasible set (Budget set), preference relation, andbehavioral Consumption Set and Budget ConstraintAssume that there areLgoods.

2 Consumption set: set of all conceivable consumption bundles =:X x= (x1, ,xL) UsuallyX=RL+but more specific sometimescf)X= bundles that gives the consumer a subsistence existence We assume thatX RL+is closed and convex Budget constraint: set of all affordable bundles given pricesp= (p1, ,pL) and incomem, which is given byB(p,m) ={x X|p x m}cf) Budget constraint with 2 Preferences and Utility Preference: binary relation onXto compare or order the bundles x y: xis at least as good asy weak preference x yiffx yandy x: xis strictly preferred toy strict preference x yiffx yandy x: xis indifferent toy indifference relation Subsets ofXderived from the preference relation: for a given bundley X, Upper contour set:P(y) ={x X|x y} Strictly upper contour set:Ps(y) ={x X|x y} Lower contour and strictly lower contour sets, denotedL(y) andLs(y) Indifference set (or curve):I(y) ={x X|x y} The preference relation onXis calledrationalif it possesses the following properties: Complete: x,y X, eitherx yory x Transitive.

3 X,y,z X, ifx yandy z, thenx z Other desirable properties Monotonicity: x,y X, ifx y, thenx ywhile ifx y, thenx y Strict monotonicity: x,y X, ifx yandx6=y, thenx y Continuity: y X,P(y) andL(y) are closed orPs(y) andLs(y) are (Lexicographic Preference)AssumeL= 2. Define as follows:x yiff either x1> y1 or x1=y1andx2 y2 Neither continuous nor evenupper semi-continuous (or the upper contour set is not closed) Non-satiation: x X, y Xsuch thaty x Local non-satiation: x Xand >0, y Xwith|x y|< such thaty x Convexity: Ifx y, thentx+ (1 t)y y, t [0,1] Strict convexity: Ifx yandx6=y, thentx+ (1 t)y y, t (0,1) The Utility real-valued functionu:X Ris called a utility function representingthe preference relation iff x,y X,u(x) u(y) x (Some Utility Functions)(1) Cobb-Douglas.

4 U(x) =x 11x 22 x LLwith >0, continuous, strictly mono-tone, and strictly convex inRL++(2) Linear:u(x) =PL =1 x continuous, strictly monotone, and convex inRL+(3) Leontief:u(x) = min{ 1x1, , LxL} continuous, monotone, and convex inRL+ preference relation can be represented by a utility function onlyif it is converse, however, does not hold. That is, a rational preference in itself doesnotguarantee the existence of utility function representing (Existence of a Utility Function)Suppose that preference relation is complete, reflexive, transitive, continuous, and strictly monotonic.

5 Then, there existsa continuous utility functionu:RL+ Rwhich represents . (1,1, ,1) RL+. Then, given any vectorx RL+, letu(x) be definedsuch thatx u(x)e. We first show thatu(x) exists and is :LetB={t R|te x}andW={t R|x te}. NeitherBnorWis emptywhileB W=R+. Also, the continuity of implies both sets are closed. Since the realline is connected, there existstx Rsuch thattxe :Ift1e xandt2e x, thent1e t2eby the transitivity of . So, by strictmonotonicity,t1= , we show thatu( ) represents , which results from the following: x1,x2 RL+,x1 x2 u(x1)e x1 x2 u(x2)e u(x1)e u(x2)e u(x1) u(x2).

6 Continuity ofu( ):Consider{xm}withxm x. Suppose to the contrary thatu(xm)9u(x). Consider the caseu := limk u(xm)> u(x). Then, by monotonicity,u e u(x) u:=12[u +u(x)]. By monotonicity, ue> u(x)e. Now, sinceu(xm) u > u, Msuchthat for allm > M,u(xm)> u. For all suchm,xm u(xm)e ue. By the continuity of , this would implyx ue, which in turn impliesu(x)e x ue, a contradiction. Theother caseu < u(x) can be dealt with (Non-representation of Lexicographic Preference by a UtilityFunction)Lexicographic preference cannot be represented by any function whether contin-uous or utility function is said to be unique up to the monotonic transformation in thefollowing (Invariance of Utility Function to Positive Monotonic Trans-forms)Ifu(x)represents some preference andf:R Ris strictly increasing, thenv(x) =f(u(x))represents the same is becausef(u(x)) f(u(y)) iffu(x) u(y).

7 The utility function inherits the properties of the preference relation that it be represented byu:RL+ R. Then, (x)is strictly increasing iff is strictly (x)is quasiconcave iff is (x)is strictly quasiconcave iff is strictly strict quasiconcavity ofu(x) can be checked by verifying if the principalminors of bordered Hessian have determinants alternating in sign: 0u1u2u1u11u12u2u21u22 >0, 0u1u2u3u1u11u12u13u2u21u22u23u3u31u32u33 <0,and so on, whereu = u x andu k= 2u x xk. Marginal rate of substitution: given a bundlex= (x1,x2) R2+,MRS12atx:= u(x)/ x1 u(x)/ x2 This measures the (absolute value of) slope of indifference curve at bundley Invariance of MRS to monotone transforms: ifv(x) =f(u(x)), then v(x)/ x1 v(x)/ x2=f (u) u(x)/ x1f (u) u(x)/ x2= u(x)/ x1 u(x)/ Utility Maximization and Optimal ChoiceA fundamental hypothesis in the consumer Theory is that a rational consumer will choosea most preferred bundle from the set of affordable alternatives.

8 Utility maximization problem: forp 0, andm >0,maxx RL+u(x) B(p,m) orp x m. At least one solution exists sinceB(p,m) is compact andu(x) is continuous (by theWeierstrass theorem). Ifu(x) is strictly quasiconcave, then the solution is unique. Ifu(x) is locally non-satiated, then the budget constraint is binding at the optimum. Marshallian demand correspondence or function:x(p,m) = arg maxx RL+u(x) x m x(tp,tm) =x(p,m): homogeneous of degree zero x(p,m) is continuous by the Berge s Maximum Theorem. Lagrangian method and first-order condition:L(x, ) =u(x) + [m p x],where 0 is Lagrangian multiplier associated with the budget constraint.

9 Ifx x(p,m), then there exists a Lagrange multiplier 0 such that for alli= 1, ,L, u(x ) x p ,with equality ifx >0. Ifx 0, then it is called interior solution: for all andk, u(x ) x = p ,so u(x )/ x u(x )/ xk=p pk7 This may not hold if the solution is not interiorMRS12(x )>p1p2 measures the change in utility from a marginal increase inm:LX =1 u(x(p,m)) x x (p,m) m=LX =1 p x (p,m) m= , : shadow price of (Demand function for the Cobb-Douglas Utility Function)Bylog transformf(u) = lnu, we havev(x) =f(u(x)) = lnx1+ (1 ) , theconsumer solvesmaxx1,x2 lnx1+ (1 ) +p2x2=m,which results in the following s x1= p11 x2= , at the optimum,p1x1= 1 p2x2, which can be substituted into the budget constraintto yieldp1x1= 1 (m p1x1).

10 Thus, we havex1(p,m) = mp1andx2(p,m) =(1 )mp2. Indirect utility functionv:RL+ R+ Ris given byv(p,m) = maxx RL+u(x) x Hence,v(p,m) =u(x(p,m)) (Properties of the Indirect Utility Function)Ifu(x)is con-tinuous and locally non-satiated onRL+and(p,m) 0, then the indirect utility functionis(1) Homogeneous of degree zero(2) Nonincreasing inpand strictly increasing inm(3) Quasiconvex inpandm.(4) Continuous (1) Follows fromv(tp,tm) =u(x(tp,tm)) =u(x(p,m)) =v(p,m).(2) Note thatB(p ,m) B(p,m) forp p. Then,v(p,m) = maxx B(p,m)u(x) maxx B(p ,m)u(x) =v(p ,m).The strict monotonicity regardingmfollows from a similar argument and the fact thatu(x) is locally non-satiated.